Write an implicit but exact formula for λn in terms of the point x = 12 and the function f (x, λ) = λx(1 −x).

Statistic Problems

Numerics of superstable cycles

Let λn denote the value of λ at which the logistic map, f (x, λ) = λx(1 −x) has a superstable cycle of period 2n.

Write an implicit but exact formula for λn in terms of the point x = 12 and the function f (x, λ) = λx(1 −x). (You will find lots of inspiration in the lecture notes, if not this very equation).

Using a computer and the result of part (a), find λ2, λ3, …, λ7 to three significant digits. Hint: formulate the problem as one where you have to find the zero of a function. Alternatively, you could make pictures, zoom in, and find those zeros by hand.

Evaluate (λ3 −λ2)/(λ4 −λ3). This will approximate the other universal constant of the period-doubling route to chaos. The problem is that success of this assignment depends a little on success of the previous one. If necessary, you could revert to (λ2 −λ1)/(λ3 −λ2). Be practical, this you would typically face in an experiment.